Freundlich empirical equation

Equation 6.9 is the same as the Freundlich empirical equation 6.1. This agreement was taken as a proof for the theory of monomolecular adsorption advanced by Langmuir. 

The mathematical models that have been applied to the physical adsorption from liquid solutions are generally extensions of the theories that have been developed to describe the sorption of gases on solid surfaces with modifications to account for the competition between the solute and solvent for the adsorption sites. Two of these models have been applied to the adsorption isotherms of nonelectrolytes from solution they are the Langmuir model and the Brunauer, Emmett, and Teller (BET) model in addition the Freundlich empirical equation has also been used. In the Langmuir model it is assumed that the adsorbed species forms a monolayer on the surface of the adsorbent, that the adsorbed molecules... 

The Freundlich empirical equation [8] has been proposed to fit adsorption data ... 

The log-log plots are insensitive within the range of experimental data, and the equation contains two empirical constants, and N. Because the Freundlich equation is empirical in nature, it provides very little insight into the mechanism regulating phosphorus adsorption process. 

Freundlich adsorption equation, although successful to explain many solution adsorption data, has failed to explain the data at very high and low concentrations. This is perhaps due to the fact that the Freundlich equation is empirical in nature and thermodynamically inconsistent at high and low concentrations. Thus, a theoretical analysis of adsorption from solution and the derivation of a suitable equation have been comparatively difQcult as both the components of the solution compete with each other for the available surface. Moreover, the thermal motion of the molecules in the liquid phase and their mutual interactions are much less well understood. It is, therefore, difQcult to correctly assess the nature of the adsorbed phase, whether monomo-lecular or multimolecular. The nature of the phase is usually determined by the nature of the carbon as well as by the nature of the components of the solution, the concentration of the solution, and the mutual solubility of the components. 

The Langmuir equation has a strong theoretical basis, whereas the Freundlich equation is an almost purely empirical formulation because the coefficient N has embedded in it a number of thermodynamic parameters that cannot easily be measured independently.120 These two nonlinear isotherm equations have most of the same problems discussed earlier in relation to the distribution-coefficient equation. All parameters except adsorbent concentration C must be held constant when measuring Freundlich isotherms, and significant changes in environmental parameters, which would be expected at different times and locations in the deep-well environment, are very likely to result in large changes in the empirical constants. 

Empirical Models vs. Mechanistic Models. Experimental data on interactions at the oxide-electrolyte interface can be represented mathematically through two different approaches (i) empirical models and (ii) mechanistic models. An empirical model is defined simply as a mathematical description of the experimental data, without any particular theoretical basis. For example, the general Freundlich isotherm is considered an empirical model by this definition. Mechanistic models refer to models based on thermodynamic concepts such as reactions described by mass action laws and material balance equations. The various surface complexation models discussed in this paper are considered mechanistic models. 

Adsorption from liquids is less well understood than adsorption from gases. In principle the equations derived for gases ought to be applicable to liquid systems, except when capillary condensation is occurring. In practice, some offer an empirical fit of the equilibrium data. One of the most popular adsorption isotherm equations used for liquids was proposed by Freundlich 21-1 in 1926. Arising from a study of the adsorption of organic compounds from aqueous solutions on to charcoal, it was shown that the data could be correlated by an equation of the form ... 

In general, there is an array of equilibrium-based mathematical models which have been used to describe adsorption on solid surfaces. These include the widely used Freundlich equation, a purely empirical model, and the Langmuir equation as discussed in the following sections. More detailed modeling approaches of sorption mechanisms are discussed in more detail in Chap. 3 of this volume. 

The Freundlich form is often employed along with the Langmuir form and is referred to as the Langmuir—FreundUch equation. That model is the basis of a great many useful modifications including many empirical forms that serve to describe adsorption data quite accurately. 

The Freundlich equation was derived empirically, based on the logarithmic decrease in adsorption energy with increasing coverage of the adsorbent snrface. Freundlich fonnd that adsorption data for many dilnte solntions conld be fit by the expression... 

The above surface complexation models enable adsorption to be related to such parameters as the number of reactive sites available on the oxide surface, the intrinsic, ionization constants for each type of surface site (see Chap. 10), the capacitance and the binding constants for the adsorbed species. They, therefore, produce adsorption isotherms with a sounder physical basis than do empirical equations such as the Freundlich equation. However, owing to differences in the choice of adjustable... 

For separation by adsorption, adsorption capacity is often the most important parameter because it determines how much adsorbent is required to accomplish a certain task. For the adsorption of a variety of antibiotics, steroids, and hormons, the adsorption isotherm relating the amount of solute bound to solid and that in solvent can be described by the empirical Freundlich equation. 

Sorption is most commonly quantified using distribution coefficients (Kd), which simplistically model the sorption process as a partitioning of the chemical between homogeneous solid and solution phases. Sorption is also commonly quantified using sorption isotherms, which allow variation in sorption intensity with triazine concentration in solution. Sorption isotherms are generally modeled using the empirical Freundlich equation, S = K CUn, in which S is the sorbed concentration after equilibration, C is the solution concentration after equilibration, and Kt and 1 In are empirical constants. Kd and K are used to compare sorption of different chemicals on one soil or sorbent, or of one chemical on several sorbents. Kd and K are also commonly used in solute leaching models to predict triazine interactions with soils under various environmental conditions. 

Geochemical models of sorption and desorption must be developed from this work and incorporated into transport models that predict radionuclide migration. A frequently used, simple sorption (or desorption) model is the empirical distribution coefficient, Kj. This quantity is simply the equilibrium concentration of sorbed radionuclide divided by the equilibrium concentration of radionuclide in solution. Values of Kd can be used to calculate a retardation factor, R, which is used in solute transport equations to predict radionuclide migration in groundwater. The calculations assume instantaneous sorption, a linear sorption isotherm, and single-valued adsorption-desorption isotherms. These assumptions have been shown to be erroneous for solute sorption in several groundwater-soil systems (1-2). A more accurate description of radionuclide sorption is an isothermal equation such as the Freundlich equation ... 

The problem of predicting multicomponent adsorption equilibria from single-component isotherm data has attracted considerable attention, and several more sophisticated approaches have been developed, including the ideal adsorbed solution theory and the vacancy solution theory. These theories provide useful quantitative correlations for a number of binary and ternary systems, although available experimental data are somewhat limited. A simpler but purely empirical approach is to use a modified form of isotherm expression based on Langmuir-Freundlich or loading ratio correlation equations ... 

The Freundlich equation, empirical in origin, relates positive adsorption to a power function of c, as follows ... 

For the intermediate concentrations, another empirical equation, the Freundlich equation, for adsorption at given temperature can be given as ... 

Freundlich,1 Rideal,2 and Hiickel3 have summarized the results of adsorption measurements on porous substances. Very roughly, it may be said that the ease of adsorption is proportional to the ease with which the gases can be liquefied, a fact which has been held to indicate that there is sometimes actual condensation of vapour in the smallest pores. Numerous empirical formulae, of which Freundlich s equation ( 9) is the best known, have been developed, but. none seem to fit the data with any accuracy over a considerable range of pressures. 

When plotted as log -SC versus log C, Equation 4.6 produces a linear plot with log ATd as the y intercept and 1/n as the slope. Figure 4.15 shows the linearized form of the Freundlich plot of the data in Figure 4.14. The linearized Freundlich plot has no particular molecular mechanistic interpretation it simply represents an empirical approach for predicting the distribution of a constituent (e.g., herbicide) between the... 

The variation of adsorption with pressure at a given constant temperature is generally expressed graphically as shown in the figure given above. Each curve is known as adsorption isotherm for a particular temperature. The relation ship between the magnitude of adsorption and pressure can be expressed mathematically by a empirically equation commonly known as Freundlich adsorption isotherm, viz. 

There is no assurance that above conclusions of the Freundlich equation is unique consequently, if data fit the equation it is only likely but not proven, that surface is heterogeneous. Basically the equation is an empirical one, limited in its usefulness to its ability to fit data. 

Many different equations have been applied to physisorption isotherms on micro porous adsorbents. The first and best known empirical equation was proposed by Freundlich (1926) in the form... 

The adsorption capacity of activated carbon may be determined by the use of an adsorption isotherm. The adsorption isotherm is an equation relating the amount of solute adsorbed onto the solid and the equilibrium concentration of the solute in solution at a given temperature. The following are isotherms that have been developed Freundlich Langmuir and Brunauer, Emmet, and Teller (BET). The most commonly used isotherm for the application of activated carbon in water and wastewater treatment are the Ereundlich and Langmuir isotherms. The Freundlich isotherm is an empirical equation the Langmuir isotherm has a rational basis as will be shown below. The respective isotherms are ... 

One of such empirical equations is that of Freundlich 11.1.21 and another one is the Temkin equation l... 

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